at is called the parametric speed. It is, by definition, non-negative(s being measured always the sense of increasing t) If the parametric speed does not vary significantly, parameter values to, t1, . tN corre- sponding to a uniform increment At= tk-tk-l, will be evenly distributed along the curve, as illustrated in Figure 2.6 罡 Parameter space Figure 2.6: When parametric speed does not vary, parameter values are uniformly spaced alon a parametric curve. The arc length of a segment of the curve between points r(to) and r(t)can be obtained as follows s(t)= (t)+y2(t)+22(t)dt Derivatives of arc length s w.r. t. parameter t and vice versa r|= 一此dddd (r·r)(i r (·r)2 (·P+r,r)(r)-4(·i)2 )2 (28 2.3 Tangent vector The vector r(t+At)-rt)indicates the direction from r(t)tor(t+At). If we divide the vector by At and take the limit as At -0, then the vector will converge to the finite magnitude vector r(t) r(t) is called the tangent vector. Magnitude of the tangent vector r dtds dt is called the parametric speed. It is, by definition, non-negative (s being measured always in the sense of increasing t). If the parametric speed does not vary significantly, parameter values t0,t1, · · · ,tN corre￾sponding to a uniform increment ∆t = tk − tk−1, will be evenly distributed along the curve, as illustrated in Figure 2.6. Parameter Space t0 t1 t2 t3 t4 t5 t0 t1 t2 t3 t4 t5 x y t ds dt Figure 2.6: When parametric speed does not vary, parameter values are uniformly spaced along a parametric curve. The arc length of a segment of the curve between points r(to) and r(t) can be obtained as follows: s(t) = Z t to q x˙ 2 (t) + y˙ 2 (t) + z˙ 2 (t)dt = Z t to √ r˙ · r˙dt (2.2) Derivatives of arc length s w.r.t. parameter t and vice versa : s˙ = ds dt = |r˙| = √ r˙ · r˙ (2.3) s¨ = ds˙ dt = r˙ · ¨r √ r˙ · r˙ (2.4) ··· s = ds¨ dt = (r˙ · r˙)(r˙· ··· r +¨r · ¨r) − (r˙ · ¨r) 2 (r˙ · r˙) 3 2 (2.5) t 0 = dt ds = 1 |r˙| = 1 √ r˙ · r˙ (2.6) t 00 = dt0 ds = − r˙ · ¨r (r˙ · r˙) 2 (2.7) t 000 = dt00 ds = − (¨r · ¨r + r˙· ··· r)(r˙ · r˙) − 4(r˙ · ¨r) 2 (r˙ · r˙) 7 2 (2.8) 2.3 Tangent vector The vector r(t+∆t)−r(t) indicates the direction from r(t) to r(t+∆t). If we divide the vector by ∆t and take the limit as ∆t → 0, then the vector will converge to the finite magnitude vector r˙(t). r˙(t) is called the tangent vector. Magnitude of the tangent vector |r˙| = ds dt (2.9) 8